On 16/03/2012 04:24, Rotwang wrote: > [...] > > Is it just that the word "set" invokes the image of a > container, and you don't accept the idea of something containing an > infinite number of things? If so then just call them something other > than sets, and use a word other than "in" for the elementhood relation. > In practice, nothing that mathematicians use sets for actually depend on > thinking of sets as "containing" things.
Perhaps I should illustrate this with an example. Let's suppose that instead of talking about sets, mathematicians spoke of "flags". Instead of saying that a number n is in a set x, they would say that a number n possessed a flag x. All of the axioms of set theory could then be translated into statements about flags; the pairing axiom (which says that for all sets x and y there is a set z whose elements are x and y, and nothing else) would become the statement that, for all flags f and g, there exists a flag h such that f possesses h, g possesses h, and no other flags possess h. The axiom of infinity would become the statement that there is a flag f which is possessed by 0, and which is possessed by n + 1 whenever it is possessed by n.
The completeness axiom of the real numbers would then have to be changed to refer to flags instead of sets: instead of saying that every non-empty set of real numbers bounded above has a least upper bound, it would say that for every flag f such that at least one real number possesses f, and such that there is a real number M which is greater than every real number that possesses f, there exists a least real number greater than or equal to every real number that possesses f. The proof I gave would go through in much the same way, except it wouldn't refer to sets containing infinitely many numbers; instead it would refer to flags that are possessed by infinitely many real numbers. But mathematically it would be the exact same proof, just using different (and more cumbersome) words to mean the same thing.
Would you have any objection to the idea of a flag being possessed by infinitely many numbers?